This interpretation matches what Kalak says in RoW:
"Regardless, I write now. Because I know they are coming for me. They got Jezrien. They’ll inevitably claim me, even here in the honorspren stronghold. And so, I’ll die. Yes, die. If you’re reading this and wondering what went wrong—why my soul evaporated soon after being claimed by the gemstone in your knife—then I name you idiot for playing with powers you only presume to understand. The bond is what keeps us alive. You sever that, and we will slowly decompose into ordinary souls—with no valid Connection to the Physical or Spiritual Realms. Capture one of us with your knives, and you won’t be left with a spren in a jar, foolish ones. You’ll be left with a being that eventually fades away into the Beyond."
It seems like the Heralds' souls are augmented by investiture, not replaced by it.
